Black Holes and Covariance in Effective Quantum Gravity

The longstanding issue of general covariance in effective models of quantum gravity is addressed, which arises when canonical quantum gravity leads to a semiclassical model described by an effective Hamiltonian constraint. In the context of spherically symmetric models, general covariance is precisely formulated into a set of equations, leading to the necessary and sufficient conditions for ensuring covariance. With the aid of these conditions, we derive the equations for the effective Hamiltonian constraint. The equations yield two candidates for effective Hamiltonian constraints dependent on a quantum parameter. The resulting quantum modified black hole spacetimes are analyzed. Our models show improvement by casting off the known limitations of previous works with similar results.

💡 Research Summary

The paper tackles the long‑standing problem of maintaining four‑dimensional general covariance in effective models derived from canonical quantum gravity. Starting from a spherically symmetric reduction of general relativity, the authors work with canonical variables ((K_1,E_1)) for the radial “dilaton” sector and ((K_2,E_2)) for the two‑dimensional gravitational sector, whose Poisson brackets are ({K_1(x),E_1(y)}=2\delta(x,y)) and ({K_2(x),E_2(y)}=\delta(x,y)). The diffeomorphism constraint (H_x) retains its classical form, while the Hamiltonian constraint (H_{\rm eff}) is left undetermined and is assumed to be first‑class. The authors postulate a modified constraint algebra in which the structure function (S) acquires a quantum correction factor (\mu): \